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12 On What Grounds What
"... Substance is the subject of our inquiry; for the principles and the causes we are seeking are those of substances. For if the universe is of the nature of a whole, substance is its first part;... —Aristotle(1984: 1688; Meta.1069a18–20) On the now dominant Quinean view, metaphysics is about what ther ..."
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Substance is the subject of our inquiry; for the principles and the causes we are seeking are those of substances. For if the universe is of the nature of a whole, substance is its first part;... —Aristotle(1984: 1688; Meta.1069a18–20) On the now dominant Quinean view, metaphysics is about what there is. Metaphysics so conceived is concerned with such questions as whether properties exist, whether meanings exist, and whether numbers exist. I will argue for the revival of a more traditional Aristotelian view, on which metaphysics is about what grounds what. Metaphysics so revived does not bother asking whether properties, meanings, and numbers exist. Of course they do! The question is whether or not they are fundamental. In §1 I will distinguish three conceptions of metaphysical structure. In §2 I will defend the Aristotelian view, coupled with a permissive line on existence. In §3 I will further develop a neoAristotelian framework, built around primitive grounding relations.
Natural Deduction for NonClassical Logics
, 1996
"... We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke m ..."
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We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.
Philosophical ìntuitions' and scepticism about judgement
 Dialectica
, 2004
"... 1. What are called ‘intuitions ’ in philosophy are just applications of our ordinary capacities for judgement. We think of them as intuitions when a special kind of scepticism about those capacities is salient. 2. Like scepticism about perception, scepticism about judgement pressures us into conceiv ..."
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1. What are called ‘intuitions ’ in philosophy are just applications of our ordinary capacities for judgement. We think of them as intuitions when a special kind of scepticism about those capacities is salient. 2. Like scepticism about perception, scepticism about judgement pressures us into conceiving our evidence as facts about our internal psychological states: here, facts about our conscious inclinations to make judgements about some topic rather than facts about the topic itself. But the pressure should be resisted, for it rests on bad epistemology: specifically, on an impossible ideal of unproblematically identifiable evidence. 3. Our resistance to scepticism about judgement is not simply epistemic conservativism, for we resist it on behalf of others as well as ourselves. A reason is needed for thinking that beliefs tend to be true. 4. Evolutionary explanations of the tendency assume what they should explain. Explanations that appeal to constraints on the determination of reference are more promising. Davidson’s truthmaximizing principle of charity is examined but rejected. 5. An alternative principle is defended on which the nature of reference is to maximize knowledge rather than truth. It is related to an externalist conception of mind on which knowing is the central mental state. 6. The knowledgemaximizing
Validity concepts in prooftheoretic semantics
 ProofTheoretic Semantics. Special issue of Synthese
"... Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in det ..."
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Abstract. The standard approach to what I call “prooftheoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of prooftheoretic semantics, this paper investigates in detail various notions of prooftheoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart. 1. Introduction: Prooftheoretic
From a Brouwerian point of view
 Philosophia Mathematica
, 1998
"... In the paper below we will discuss a number of topics that are central in Brouwer’s intuitionism. A complete treatment is beyond the scope of the paper, the reader may find it a useful introduction to Brouwer’s papers. There are a number of loosely related notions and schools of constructivism in m ..."
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In the paper below we will discuss a number of topics that are central in Brouwer’s intuitionism. A complete treatment is beyond the scope of the paper, the reader may find it a useful introduction to Brouwer’s papers. There are a number of loosely related notions and schools of constructivism in mathematics; in some cases there is only an attempt to capture certain constructive notions and procedures in the existing body of mathematics, in other, more fundamental, cases the object is to reconstruct mathematics as a whole within the frame work of a constructivistic philosophy. It is almost a platitude to state that constructivism has always been around in mathematics, and indeed, a, say, eighteenth century mathematician would have accepted the constructivist claim of his twentieth century colleague of the mild variety, i.e. the practical nondogmatic practitioner, as selfevident and rather commonplace. The issue of constructivism in mathematics only became urgent after the discovery of abstract, noneffective techniques and notions. The watershed is David Hilbert’s famous solution
Epistemic truth and excluded middle*
"... Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemi ..."
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Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution. Part I The Problem §1. The epistemic conception of truth. The epistemic conception of truth can be formulated in many ways. But the basic idea is that truth is explained in terms of epistemic notions, like experience, argument, proof, knowledge, etc. One way of formulating this idea is by saying that truth and knowability coincide, i.e. for every statement S
unknown title
"... It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is ..."
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It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is sketched, intended to provide a coherent methodological stance towards the issue. Some reasons to recommend this stance are given, as well as some speculations as to why not everyone might want to follow the recommendation. 1.
Why Husserl should have been a strong revisionist in mathematics ∗
, 2000
"... Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’ ..."
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Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’s third claim is wrong, by his own standards. To explain this thesis, let me first introduce the term ‘revisionism’. It is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice ’ [Wright 1980, p.117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in different directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. ‘May lead’, not ‘necessarily leads’: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of nonrevisionism, weak revisionism, and strong revisionism. Nonrevisionism can be found in Wittgenstein’s Philosophische Untersuchungen, where philosophy can neither change nor ground mathematics: Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische